Move the origin point to the cursor in Edit Mode
What I understood  about your issue, you want to open mouth and close it. 

Please try following steps:


Select the base edges as shown in the image below, snap the 3D cursor to this edge, by hitting Shift + S and selecting Cursor to selected

Change the Pivot Point to 3D Cursor

Select the full mouth and rotate it as required.



You may also create a Shape key for easy rotate and animate

When you are done with it you may change back the Pivot point to Median point

Best of luck!